Theorem opnnei 23112

Description: A set is open iff it is a neighborhood of all of its points. (Contributed by Jeff Hankins, 15-Sep-2009.)

Assertion

Ref	Expression
opnnei	⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))

Distinct variable groups:   𝑥,𝐽   𝑥,𝑆

Proof of Theorem opnnei

Step	Hyp	Ref	Expression
1		0opn 22894	. . . . 5 ⊢ (𝐽 ∈ Top → ∅ ∈ 𝐽)
2	1	adantr 479	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∅ ∈ 𝐽)
3		eleq1 2814	. . . . 5 ⊢ (𝑆 = ∅ → (𝑆 ∈ 𝐽 ↔ ∅ ∈ 𝐽))
4	3	adantl 480	. . . 4 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆 ∈ 𝐽 ↔ ∅ ∈ 𝐽))
5	2, 4	mpbird 256	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → 𝑆 ∈ 𝐽)
6		rzal 4503	. . . 4 ⊢ (𝑆 = ∅ → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
7	6	adantl 480	. . 3 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
8	5, 7	2thd 264	. 2 ⊢ ((𝐽 ∈ Top ∧ 𝑆 = ∅) → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
9		opnneip 23111	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽 ∧ 𝑥 ∈ 𝑆) → 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
10	9	3expia 1118	. . . . . 6 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → (𝑥 ∈ 𝑆 → 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
11	10	ralrimiv 3135	. . . . 5 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ 𝐽) → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
12	11	ex 411	. . . 4 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
13	12	adantr 479	. . 3 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆 ∈ 𝐽 → ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
14		df-ne 2931	. . . . . 6 ⊢ (𝑆 ≠ ∅ ↔ ¬ 𝑆 = ∅)
15		r19.2z 4489	. . . . . . 7 ⊢ ((𝑆 ≠ ∅ ∧ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}))
16	15	ex 411	. . . . . 6 ⊢ (𝑆 ≠ ∅ → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
17	14, 16	sylbir 234	. . . . 5 ⊢ (¬ 𝑆 = ∅ → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → ∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
18		eqid 2726	. . . . . . . 8 ⊢ ∪ 𝐽 = ∪ 𝐽
19	18	neii1 23098	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑆 ⊆ ∪ 𝐽)
20	19	ex 411	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ⊆ ∪ 𝐽))
21	20	rexlimdvw 3150	. . . . 5 ⊢ (𝐽 ∈ Top → (∃𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ⊆ ∪ 𝐽))
22	17, 21	sylan9r 507	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ⊆ ∪ 𝐽))
23	18	ntrss2 23049	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
24	23	adantr 479	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) ⊆ 𝑆)
25		vex 3466	. . . . . . . . . . . . 13 ⊢ 𝑥 ∈ V
26	25	snss 4784	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆))
27	26	ralbii 3083	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆))
28		dfss3 3967	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ ((int‘𝐽)‘𝑆) ↔ ∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆))
29	28	biimpri 227	. . . . . . . . . . . 12 ⊢ (∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
30	29	adantl 480	. . . . . . . . . . 11 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 𝑥 ∈ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
31	27, 30	sylan2br 593	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → 𝑆 ⊆ ((int‘𝐽)‘𝑆))
32	24, 31	eqssd 3996	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)) → ((int‘𝐽)‘𝑆) = 𝑆)
33	32	ex 411	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆) → ((int‘𝐽)‘𝑆) = 𝑆))
34	25	snss 4784	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑆 ↔ {𝑥} ⊆ 𝑆)
35		sstr2 3985	. . . . . . . . . . . . . 14 ⊢ ({𝑥} ⊆ 𝑆 → (𝑆 ⊆ ∪ 𝐽 → {𝑥} ⊆ ∪ 𝐽))
36	35	com12 32	. . . . . . . . . . . . 13 ⊢ (𝑆 ⊆ ∪ 𝐽 → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ ∪ 𝐽))
37	36	adantl 480	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → ({𝑥} ⊆ 𝑆 → {𝑥} ⊆ ∪ 𝐽))
38	34, 37	biimtrid 241	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑥 ∈ 𝑆 → {𝑥} ⊆ ∪ 𝐽))
39	38	imp 405	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ 𝑆) → {𝑥} ⊆ ∪ 𝐽)
40	18	neiint 23096	. . . . . . . . . . . 12 ⊢ ((𝐽 ∈ Top ∧ {𝑥} ⊆ ∪ 𝐽 ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
41	40	3com23 1123	. . . . . . . . . . 11 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽 ∧ {𝑥} ⊆ ∪ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
42	41	3expa 1115	. . . . . . . . . 10 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ {𝑥} ⊆ ∪ 𝐽) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
43	39, 42	syldan 589	. . . . . . . . 9 ⊢ (((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) ∧ 𝑥 ∈ 𝑆) → (𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
44	43	ralbidva 3166	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) ↔ ∀𝑥 ∈ 𝑆 {𝑥} ⊆ ((int‘𝐽)‘𝑆)))
45	18	isopn3 23058	. . . . . . . 8 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (𝑆 ∈ 𝐽 ↔ ((int‘𝐽)‘𝑆) = 𝑆))
46	33, 44, 45	3imtr4d 293	. . . . . . 7 ⊢ ((𝐽 ∈ Top ∧ 𝑆 ⊆ ∪ 𝐽) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ∈ 𝐽))
47	46	ex 411	. . . . . 6 ⊢ (𝐽 ∈ Top → (𝑆 ⊆ ∪ 𝐽 → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ∈ 𝐽)))
48	47	com23 86	. . . . 5 ⊢ (𝐽 ∈ Top → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 ⊆ ∪ 𝐽 → 𝑆 ∈ 𝐽)))
49	48	adantr 479	. . . 4 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → (𝑆 ⊆ ∪ 𝐽 → 𝑆 ∈ 𝐽)))
50	22, 49	mpdd 43	. . 3 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥}) → 𝑆 ∈ 𝐽))
51	13, 50	impbid 211	. 2 ⊢ ((𝐽 ∈ Top ∧ ¬ 𝑆 = ∅) → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))
52	8, 51	pm2.61dan 811	1 ⊢ (𝐽 ∈ Top → (𝑆 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑆 𝑆 ∈ ((nei‘𝐽)‘{𝑥})))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   = wceq 1534   ∈ wcel 2099   ≠ wne 2930  ∀wral 3051  ∃wrex 3060   ⊆ wss 3946  ∅c0 4322  {csn 4623  ∪ cuni 4905  ‘cfv 6546  Topctop 22883  intcnt 23009  neicnei 23089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-id 5572  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-top 22884  df-ntr 23012  df-nei 23090

This theorem is referenced by:  neiptopreu  23125  flimcf  23974